Listen to Robert Emmerich introduce The Big Apple, a hit song from 1937. Music written by Bob and performed by Tommy Dorsey's Clambake Seven with Bob on piano. Lyrics written by Buddy Bernier and sung by Edythe Wright. Audio provided by Dorothy Emmerich.

The anecdote might be apocryphal; it first appears in print in the New York (NY) Post in 1964. Later versions replaced Jimmy Walker with New York politician Al Smith (1873-1944).

Early versions of the joke appeared in 1953 and 1957, without reference to any specific people.

20 December 1953, Los Angeles (CA) Times, “Fact and Comment” by Charles C. Cohan, pt. 5, pg. 3, col. 1:
A westerner visiting in Maine, or maybe Vermont, asked an old-timer there, “Why do you people around here always answer a question with a question?” To which the other replied quietly, “Do we?”

18 October 1957, Washington (DC) Post and Times Herald, “The District Line” by Bill Gold, pg. B14, cols. 3-4:
FAVORITE STORY
I love the one about the Dutchman who says to his Irish friend: “You Irish have one trait I just can’t abide. Tell me, why do you always answer a question with another question?”

Old Fulton NY Post Cards
13 March 1964, New York (NY) Post, “It Happened Last Night” by Earl Wilson, pg. 20, col. 2:
“Do you know,” somebody asked the late James J. Walker, “the Irish always answer a question with a question?"..."Do they now?” asked Jimmy.

10 December 1967, “Charlie Rice’s Punchbowl,” Parade magazine, pg. 14, col. 2:
Jovial John Kieran, who was once described as “half-sportswriter, half-naturalist and half-Irish,” loved to argue ornithological points with his friends. One day he was in a prolonged discussion with Fairfield Osborne, head of the New York Zoological Society, over the migratory habits of whooping cranes. Osborne finally threw up his hands is despair. “I can’t argue with you, John—why do you Irishmen always answer a question with another question?”

Said John in a tone of injured surprise, “Do we?”

15 March 1968, Springfield (MA) Union, “New York Scene: Dishing Out the Blarney” by Norton Mockridge, pg. 17, col. 3:
Anyway, John McCarthy favorite Irish anecdote features FDR and the late James J. Walker. when Walker was mayor of New York, Roosevelt asked him:

“Why do you Irish always answer a question with a question?”

“Do we now?” replied Jimmy.

Google BooksAmerica Is Also Irish
By Robert N Webb
New York, NY: G.P. Putnam’s Sons
1973
Pg. 107:
The Irish-American was long noted for replying to a question with a question. There is a story that President Franklin D. Roosevelt, when governor of New York, once asked Mayor Jimmy Walker why the Irish always answered with a question.

“And do we, now?” the irrepressible Jimmy replied.

17 March 1977, Chicago (IL) Tribune, “In the Wake of the News” by David Condon, sec. 4, pg. 5, col. 2:
James J. Walker, the mayor of New York who could take it and get away with it, and who, when asked why the Irish always answer a question with a question, responded, “Dow we, now?”

6 March 1984, New York (NY) Times, “Tidbits From Roasting of O’Neill,” pg. B8, col. 5:
The famed Irish Mayor of New York, Jimmy Walker, had a considerable reputation for being as quick with his tongue as he was nimble on his feet. Walker was once asked by Franklin Delano Roosevelt, then the Governor, “Why is it that the Irish always answer a question with a question?”

“Do we now?” responded Walker.

Google BooksHow to Be Irish: (Even If You Already Are)
By Sean Kelly and Rosemary Rogers
New York, NY: Villard Books
1999
Pg. 32:
FDR is supposed to have asked Al Smith, “Why do you Irish always answer a question with a question?” To which the Happy Warrior replied, “Do we, now?”

Google BooksCan You Say a Few Words?:
How to Prepare and Deliver a Speech for Any Special Occasion (Revised Edition)
By Joan Detz
New York, NY: St. Martin’s Griffin
2006
Pg. 164:
The famed mayor of New York City, Jimmy Walker, who was known for being quick-witted, was once asked by then governor of New York Franklin Delano Roosevelt, “Why is it that the Irish always answer a question with a question?”

“Do we now?” responded Hizzoner.

iget2work.com
March 15, 2010
St Patrick’s Day Jokes and One Liners
(...)
Why do you Irish always answer a question with a question?’ asked President Franklin D. Roosevelt. ’Do we now?’ came New York Mayor Al Smith’s reply.